3.210 \(\int (1+2 x) \sqrt{2-x+3 x^2} (1+3 x+4 x^2) \, dx\)

Optimal. Leaf size=93 \[ \frac{2}{15} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2+\frac{(738 x+745) \left (3 x^2-x+2\right )^{3/2}}{1620}+\frac{19 (1-6 x) \sqrt{3 x^2-x+2}}{2592}+\frac{437 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{5184 \sqrt{3}} \]

[Out]

(19*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/2592 + (2*(1 + 2*x)^2*(2 - x + 3*x^2)^(3/2))/15 + ((745 + 738*x)*(2 - x + 3
*x^2)^(3/2))/1620 + (437*ArcSinh[(1 - 6*x)/Sqrt[23]])/(5184*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0657437, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1653, 779, 612, 619, 215} \[ \frac{2}{15} \left (3 x^2-x+2\right )^{3/2} (2 x+1)^2+\frac{(738 x+745) \left (3 x^2-x+2\right )^{3/2}}{1620}+\frac{19 (1-6 x) \sqrt{3 x^2-x+2}}{2592}+\frac{437 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{5184 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x)*Sqrt[2 - x + 3*x^2]*(1 + 3*x + 4*x^2),x]

[Out]

(19*(1 - 6*x)*Sqrt[2 - x + 3*x^2])/2592 + (2*(1 + 2*x)^2*(2 - x + 3*x^2)^(3/2))/15 + ((745 + 738*x)*(2 - x + 3
*x^2)^(3/2))/1620 + (437*ArcSinh[(1 - 6*x)/Sqrt[23]])/(5184*Sqrt[3])

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int (1+2 x) \sqrt{2-x+3 x^2} \left (1+3 x+4 x^2\right ) \, dx &=\frac{2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{1}{60} \int (1+2 x) (8+164 x) \sqrt{2-x+3 x^2} \, dx\\ &=\frac{2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{(745+738 x) \left (2-x+3 x^2\right )^{3/2}}{1620}-\frac{19}{216} \int \sqrt{2-x+3 x^2} \, dx\\ &=\frac{19 (1-6 x) \sqrt{2-x+3 x^2}}{2592}+\frac{2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{(745+738 x) \left (2-x+3 x^2\right )^{3/2}}{1620}-\frac{437 \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx}{5184}\\ &=\frac{19 (1-6 x) \sqrt{2-x+3 x^2}}{2592}+\frac{2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{(745+738 x) \left (2-x+3 x^2\right )^{3/2}}{1620}-\frac{\left (19 \sqrt{\frac{23}{3}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{5184}\\ &=\frac{19 (1-6 x) \sqrt{2-x+3 x^2}}{2592}+\frac{2}{15} (1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}+\frac{(745+738 x) \left (2-x+3 x^2\right )^{3/2}}{1620}+\frac{437 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{5184 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0286907, size = 60, normalized size = 0.65 \[ \frac{6 \sqrt{3 x^2-x+2} \left (20736 x^4+31536 x^3+24072 x^2+17374 x+15471\right )-2185 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{77760} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x)*Sqrt[2 - x + 3*x^2]*(1 + 3*x + 4*x^2),x]

[Out]

(6*Sqrt[2 - x + 3*x^2]*(15471 + 17374*x + 24072*x^2 + 31536*x^3 + 20736*x^4) - 2185*Sqrt[3]*ArcSinh[(-1 + 6*x)
/Sqrt[23]])/77760

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 81, normalized size = 0.9 \begin{align*}{\frac{8\,{x}^{2}}{15} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{89\,x}{90} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{961}{1620} \left ( 3\,{x}^{2}-x+2 \right ) ^{{\frac{3}{2}}}}-{\frac{-19+114\,x}{2592}\sqrt{3\,{x}^{2}-x+2}}-{\frac{437\,\sqrt{3}}{15552}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x)*(4*x^2+3*x+1)*(3*x^2-x+2)^(1/2),x)

[Out]

8/15*x^2*(3*x^2-x+2)^(3/2)+89/90*x*(3*x^2-x+2)^(3/2)+961/1620*(3*x^2-x+2)^(3/2)-19/2592*(-1+6*x)*(3*x^2-x+2)^(
1/2)-437/15552*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))

________________________________________________________________________________________

Maxima [A]  time = 1.53582, size = 124, normalized size = 1.33 \begin{align*} \frac{8}{15} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x^{2} + \frac{89}{90} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} x + \frac{961}{1620} \,{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}} - \frac{19}{432} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{437}{15552} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) + \frac{19}{2592} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)*(3*x^2-x+2)^(1/2),x, algorithm="maxima")

[Out]

8/15*(3*x^2 - x + 2)^(3/2)*x^2 + 89/90*(3*x^2 - x + 2)^(3/2)*x + 961/1620*(3*x^2 - x + 2)^(3/2) - 19/432*sqrt(
3*x^2 - x + 2)*x - 437/15552*sqrt(3)*arcsinh(1/23*sqrt(23)*(6*x - 1)) + 19/2592*sqrt(3*x^2 - x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.62988, size = 228, normalized size = 2.45 \begin{align*} \frac{1}{12960} \,{\left (20736 \, x^{4} + 31536 \, x^{3} + 24072 \, x^{2} + 17374 \, x + 15471\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{437}{31104} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)*(3*x^2-x+2)^(1/2),x, algorithm="fricas")

[Out]

1/12960*(20736*x^4 + 31536*x^3 + 24072*x^2 + 17374*x + 15471)*sqrt(3*x^2 - x + 2) + 437/31104*sqrt(3)*log(4*sq
rt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x + 1\right ) \sqrt{3 x^{2} - x + 2} \left (4 x^{2} + 3 x + 1\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x**2+3*x+1)*(3*x**2-x+2)**(1/2),x)

[Out]

Integral((2*x + 1)*sqrt(3*x**2 - x + 2)*(4*x**2 + 3*x + 1), x)

________________________________________________________________________________________

Giac [A]  time = 1.16218, size = 92, normalized size = 0.99 \begin{align*} \frac{1}{12960} \,{\left (2 \,{\left (12 \,{\left (18 \,{\left (48 \, x + 73\right )} x + 1003\right )} x + 8687\right )} x + 15471\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{437}{15552} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)*(3*x^2-x+2)^(1/2),x, algorithm="giac")

[Out]

1/12960*(2*(12*(18*(48*x + 73)*x + 1003)*x + 8687)*x + 15471)*sqrt(3*x^2 - x + 2) + 437/15552*sqrt(3)*log(-2*s
qrt(3)*(sqrt(3)*x - sqrt(3*x^2 - x + 2)) + 1)